A comparative analysis of well renowned "Shooting Method" with another numerical method "Complementary Functions Method" (CFM) is presented for calculating eigenvalue (λ). Contrary to the shooting method hit and trial approach, CFM exploits the properties of linear ordinary differential equation (LODE). In the case of linear eigenvalue Boundary value Problem (BVP), CFM generates an algebraic equation system with one unknown "λ" and, alone root finding method is sufficient to give required eigenvalue. However, the Shooting Method create a system of algebraic equations containing two unknowns "λ" and "missing initial conditions", that demands an additional numerical technique along with root finding method. These radical differences between two approaches, sets the basis for this comparative investigation. As a case study in Linear Elastic Stability, different cases of Euler columns are investigated by finding eigenvalues for each case numerically, under both methods. Comparison is performed on the basis of results accuracy and cost effectiveness for both numerical techniques while solving linear stability problems.